3. Modelling of oscillating body wave energy
converters
3.1. Introduction
Oscillating bodies are a major class of wave energy converters. The wave energy
absorption results from the interaction between the oscillating body and the incoming
waves. The device may be a single rigid body, or may be a system of bodies that can
move with respect to each other with constraints that allow relative rotation (by hinges
or similar mechanisms) or relative translation (sliding mechanisms).
A single floating oscillating body has in general six degrees of freedom: three
translations and three rotations. For a ship-like elongated body (directed parallel to the
x-axis as indicated in Fig. 3.1) the modes are named and numbered as shown in Fig. 3.1.
For an axisymmetric body or another non-elongated body, the modes 1-surge, 2-sway,
4-roll and 5-pitch are ambiguous. However the ambiguity may be removed when there
is an incident wave where the propagation direction defines the x-direction.
z
yaw 6
heave 3
y
x
Fig. 3.1. Modes of oscillation of a rigid body.
A slack-moored (as opposed to tight-moored) single body is free to oscillate in the
six modes. In other cases, the body may be hinged with respect of a fixed structure (sea
bottom, breakwater or other) and the only mode of motion is angular oscillation (pitch
or roll, depending on the geometry and on convention). This is the case of the Oyster
and the WaveRoller (both hinged at the sea bottom) and of the Wave Star and the
Brazilian hyperbaric device (whose hinges are above sea water level). In the
Archimedes Wave Swing (AWS) the only mode of motion of the active body is heave.
Sometimes, although the body may move in several degrees of freedom, the power takeoff mechanism (PTO) is associated with a particular mode of motion (frequently heave),
the other modes playing a secondary role.
In the case of multi-body devices, one body is connected to another one by a
mechanism that allows one (rarely more than one) mode of relative motion: rotation (as
in Pelamis) or translation (as in Power Buoy and WaveBob). For example, the two-body
WaveBob has 7 degrees of freedom, corresponding to the three translations and three
rotations of the pair, plus the translational motion of one body with respect to the other.
The wave energy utilization involves “large bodies” oscillating in water. It is
important to clarify what is meant here by a large body1. There are at least three relevant
1
This paragraph is based on [3.1].
1
length scales in wave-body interaction: the characteristic body dimension a, the
wavelength   2 k , and the wave amplitude Aw . Among these scales, two ratios
may be formed, for example, ka and Aw a . If the characteristic body dimension is of
the order of  2 or larger, ( ka  O(1) ), the body is regarded as large; its presence
alters the pattern of wave propagation significantly and produces diffraction. Most ships
fall into this category. For small bodies (ka  1) , such as the structural members of a
drilling tower, diffraction is of minor importance. When Aw a is sufficiently large, the
local velocity gradient near the small body augments the effect of viscosity and induces
flow separation and vortex shedding, leading to the so-called form drag. At present, the
inviscid linearized diffraction theory has been fairly well developed for Aw a  1 and
ka  O(1) with considerable experimental confirmation. The case of Aw a  O(1) and
ka  1 has been the subject of intensive experimental studies, but is not easily
describable on purely theoretical ground. The intermediate case of Aw a  O(1) and
ka  O(1) involves both separation and nonlinear diffraction and is the most difficult
and least explored area of all.
The theory that we will be developing here for oscillating-body wave energy
converters, based on inviscid potential flow, assumes that Aw a  1 and ka  O(1) , i.e.
the wave amplitude Aw is much smaller that the body characteristic length a, and a is of
the order of magnitude of  2 . In most cases, we will deal with bodies whose
characteristic length a ranges between 10 m and 50 m, and wave amplitudes not
exceeding about 2 m.
The book by Falnes [3.2] is the most complete text on the linear theory of wave
energy absorption by oscillating bodies.
3.2. Wave field of a single heaving body
We start by analyzing the simple case of a single body with a single degree of
freedom: heave oscillations (Fig. 3.2). The vertical position of the body is defined by a
vertical coordinate  from a point O fixed to the body (O may coincide or not with the
centre of gravity), with   0 in the absence of waves. The body is connected to the sea
bottom through a power take-off system (PTO) that converts the body motion into
useful energy. We denote by S the wetted surface of the body separating it from the
water, and by n the unit vector perpendicular to S pointing into the water.
The body is subject to its weight mg , where m is its mass. In the absence of waves,
in equilibrium conditions, the body weight is balanced by the upward hydrostatic force,
i.e., is equal to the weight of the displaced volume of water. So, in the dynamic
equations, we may omit the body weight and consider only disturbances (assumed small
in linear wave theory) to the pressure forces on the wetted surface of the body.
We start by considering the body fixed at its equilibrium position   0 , and assume
an incident wave whose velocity potential is i , progressing from left to right in the
positive x-direction. The presence of the body produces diffraction and disturbs the
incident wave field. Since Laplace’s equation is linear, we may construct a solution to
this problem by linear superposition, and introduce a diffraction wave field with
2
velocity potential d that, like the incident wave potential i , has to satisfy the
linearized boundary condition (2.17) on the undisturbed free surface

x
S
n
O
PTO
Fig. 3.2. Oscillating body with single degree of freedom (heave).
 2d
d
on z  0
(3.1)
z
t
and the impermeability condition d n  0 on the see bottom. The velocity
component normal to the body wetted surface has to vanish on S, which we write as
d

  i on S.
(3.2)
n
n
The excitation force results from the excess pressure pe on the body’s wetted surface.
Its vertical component may be written as
(3.3)
f e   nz pe dS ,
2
 g
S
where n z is the vertical component of the unit vector n and pe is given, from by Eq.
(2.12), by
(i  d )
.
(3.4)
pe   
t
If the diffraction term d is neglected in Eq. (3.3), the resulting force is called the
Froude-Krylov force. It may represent a reasonable approximation to the excitation
force, in particular if the extension of the immersed part of the body is very small
compared with the wavelength. It may be computationally convenient to use such an
approximation because it is not then required to solve the boundary-value problem for
finding the diffraction potential d .
We study now the force on the body when it oscillates about its equilibrium position
in the absence of any incident wave. Its vertical coordinate  is an oscillating function
of time t. The body motion produces a radiated wave field with a radiated velocity
3
potential r that is required to satisfy the linearized boundary condition (2.17) on the
undisturbed free surface
 2r

(3.5)
  g r on z  0
2
z
t
and the impermeability condition r n  0 on the see bottom. The flow velocity
relative to the body is r  (d dt )k , where k is unit vector pointing vertically
upwards and (d dt )k is the body velocity vector. On the wetted surface of the moving
body, the impermeability condition requires the normal component of this relative
velocity be zero, i.e., r n  (d dt )nz , where nz  n  k is the vertical component of
the unit normal vector n . In the present case of a heaving body, it is r z  d dt on
the instantaneous wetted surface S. In linear wave theory, we may apply this condition
at the equilibrium position of the body, it being assumed that the displacement  is a
small quantity compared with the wavelength. This is particularly convenient since the
motion of the body is in general unknown a priori. We then write
r n  (d dt )nz on the undisturbed wetted surface S.
(3.6)
The body motion produces on its surface a radiation force whose vertical component is
given by
(3.7)
f r   nz pr dS ,
S
where
r
.
(3.8)
t
Finally, if, in the absence of incident wave, the body is fixed at a position different
from it equilibrium position, i.e. if   0 , its buoyancy force no longer balances its
weight, and we have a static disturbance force f st equal to the weight of the displaced
water volume at   0 minus the corresponding value at   0 . We assume the
displacement  to be small and write approximately
f st   gScs ,
(3.9)
pr   
where S cs is the cross sectional area of the body at its position   0 by the undisturbed
horizontal free surface z  0 . If the body in the vicinity of the free surface is cylindrical
(with vertical or inclined axis), then the disturbance f st to the buoyancy force is given
exactly by Eq. (3.9).
The governing equation for the body motion is simply Newton’s second law
d 2
m 2  f e  f r  f st  f PTO ,
(3.10)
dt
where f PTO is the vertical force of the power take-off system (PTO) upon the body.
3.3. Frequency-domain analysis of wave energy absorption by a single
heaving body
3.3.1. Linear systems
The wave energy converter represented in Fig. 3.2 may be regarded as a system
whose input is the incident wave represented by the surface elevation  (t ) at a given
4
observation point (x,y), and the output is the body displacement  (t )  H  (t ). The
system is linear if, for any two inputs  1 (t ) and  2 (t ) , and for any constants c1 and c2
, it is
H c1 1  c2 2   c1H  1   c2 H  2 .
(3.11)
Linear systems are particularly simple to study theoretically. In particular, if the input
 (t ) is a simple harmonic (or sinusoidal) function of time, the output  (t )  H  (t ) is
also a simple harmonic function of time.
With the linearizing assumptions that were introduced above in the modelling of our
wave energy converter including in the boundary conditions, the converter may be
considered as a linear system, provided that the power take-off system (PTO) is also a
linear mechanism. This is the case if it consists of a linear damper and a linear spring, as
represented in Fig. 3.3. The spring may account for the presence of a mooring force.
The vertical force applied by the PTO on the floater is

x
O
C
K
Fig. 3.3. Heaving body with linear PTO consisting of a linear damper and a linear
spring.
d
 K ,
(3.12)
dt
where C is the damping coefficient and K is the spring stiffness. We assume here that
the spring force is zero at the rest position   0 .
f PTO  C
3.3.2. Governing equations and hydrodynamic coefficients
We consider an incident wave of frequency  and amplitude Aw , propagating in the
positive x-direction (from left to right) on deep water or on water of arbitrary but
uniform depth. Its potential is (see section 2)
i  i ( z) exp i(t  k x) .
(3.13)
5
Since the system is linear and the input (incident wave) is represented by a harmonic
function of time, the body displacement and the forces upon the body are also harmonic
functions of time. We may write
 (t )  X ei t ,
fe (t )  Fe ei t , f r (t )  Fr ei t ,
(3.14)
where X is the complex amplitude of the body displacement  , and Fe and Fr are the
excitation force and radiation force amplitudes respectively. These amplitudes are in
general complex. We recall that, whenever a complex expression is equated to a
physical quantity, its real part is to be taken.
The governing equation (3.10) for the body motion becomes
  2m Xei t  Fe  Fr   gScs X  iCX  KX ei t .
(3.15)
or, more simply,
  2mX  Fr   gScs X  iCX  KX  Fe .
(3.16)
In this way, we got rid of the dependence on time and are left only with (in general
complex) amplitudes. Note that Eqs (3.15) and (3.16) are valid only after the transients
associated to the initial conditions (initial position and velocity of the body) have died
out.
It is convenient to decompose the radiation force coefficient Fr as
Fr  ( 2 A  i B) X ,
(3.17)
where A and B are real coefficients. Equation (3.16) may now be rewritten as
  2 (m  A)  i ( B  C )  (  gScs  K ) X  Fe .
(3.18)
We recall that, in Eq. (3.18), Fe is the complex amplitude of the excitation force and
represents the forcing term. The real coefficient A is added to the body mass m and is
called added mass. It represents the inertia of the water surrounding the body that is
entrained by the body in its motion. The added mass A is a function of frequency  and
is in general positive, although special cases are known where the depth of submergence
is small and A may be negative (see [3.3]). The real coefficient B is added to the
damping coefficient C of the PTO and is named radiation damping coefficient (or
radiation resistance or added damping coefficient, see [3.2]).
Let us examine in more detail the radiation force. In the absence of incident waves, if
the body is forced to perform an oscillating motion  (t )  Re( X ei t ), it becomes
subject to a radiation force
f r (t )  Re( Fr ei t )  Re ( 2 A  i B) X ei t .
(3.19)
The instantaneous rate of work (force times velocity) done by this radiation force on the
moving body is Wr (t )  f r (t ) d dt . If we write X  X ei , where  is the argument
of X , we find
f r (t )  X  2 A cos(t   )   B sin(t   )
(3.20)
and
d
  X  sin( t   ) .
(3.21)
dt
We obtain
2
2
Wr (t )   1 A X  3 sin(2t  2 )  B X  2 sin 2 (t   ) .
(3.22)






2
Let us find the time-averaged value W r of Wr (t ) . (We use a bar to denote time average
over a wave period.) The contribution from the first term on the right-hand side of Eq.
6
(3.22) is obviously zero, whereas, in the second term, the average value of the squared
sinus is 1 2 . We obtain
2
(3.23)
Wr   12  2 B X .
As should be expected, the first term on the right-hand side of Eq. (3.21), representing
work done by an inertia force, does not contribute to the net work. The only non-zero
contribution to net work Wr done by the radiation force on the body comes from the
radiation force. It cannot be positive, otherwise we would be absorbing energy from
non-existing incident waves. We conclude that the radiation damping coefficient B
cannot be negative. In general it is B  0 , although special geometries have been
derived theoretically for which it is B  0 at a given frequency.
Not unexpectedly, it turns out that the coefficients A (added mass), B (radiation
damping) and   Fe Aw (excitation force amplitude per unit incident wave
amplitude) are related to each other.
We consider for a moment the general case in which the direction of the incident
wave propagation makes an angle  (     ) with the x-axis, and denote by
( )  Fe (  ) Aw the corresponding value of the excitation force amplitude per unit
incident wave amplitude. It may be shown [3.2] that, for fixed  , the following
relationship (Haskind relation) exists between B and (  )

k
( )2 d ,
B
(3.24)

2


4  g D(kh)
where D(kh) is given by any of the expressions (2.67). In the deep water limit, it is
kh   , D(kh)  1 and
3 
(3.25)
( )2 d .
3 
4  g
The added mass A( ) and the radiation damping coefficient B( ) are related to
each other by the Kramers-Kronig relations (see [3.2])
2   B( y )
A( )  A()  
dy ,
(3.26)
 0 2  y2
B
B( ) 
2 2

0
A( y )  A()
(3.27)
dy .

 2  y2
The added mass A, the radiation damping coefficient B and the excitation force
coefficient Fe Aw per unit incident wave amplitude depend on the wave frequency 
and on body geometry. Analytical expressions for these coefficients in terms of
elementary functions can be found only for some very simple geometries, like the
sphere and the horizontal-axis circular cylinder. Commercial codes, based on the
boundary-element method, are available to compute these coefficients for arbitrary
geometries and frequencies, some of the best known being WAMIT, ANSYS/Aqwa and
Aquaplus.
3.3.3. Absorbed power and power output
The instantaneous power Pabs absorbed from the waves is the instantaneous force on
the body wetted surface multiplied the body velocity d dt
7
d
,
dt
done by the PTO force is
Pabs (t )   fe (t )  f r (t )   gScs 
whereas the instantaneous rate of work PPTO
(3.28)
 d
 d
.
(3.29)
PPTO (t )  C
 K 
 dt
 dt
As we are neglecting viscous losses and other losses in water and in the PTO, the
difference between Pabs and PPTO is the rate of variation in the energy stored in the
body as kinetic energy and in the PTO spring as elastic energy. Obviously in time
average this difference vanishes and we have Pabs  PPTO  P (say). We find
1
2
P  PPTO   2C X .
(3.30)
2
or equivalently (see [3.4])
2
1
F
2 B
(3.31)
P  Pabs 
Fe  U  e ,
8B
2
2B
where U  iX is the complex amplitude of the body velocity.
Let us fix the incident wave frequency  and amplitude Aw , as well as the body
geometry (but not the PTO coefficients C and K). This implies that, in Eq. (3.31), the
values of the radiation damping coefficient B (real) and the excitation force amplitude
Fe (in general complex) are fixed. On the right-hand side of Eq. (3.31) only the
complex amplitude U  iX of the body velocity d dt is allowed to vary, since it
depends on the PTO parameters C (damper) and K (spring). We look for the conditions
that maximize the time-averaged power output P . Equation (3.31) shows that, since B
and Fe are fixed, the power output P is maximum when
F
U  i X  e .
(3.32)
2B
This condition shows that, under optimal conditions, the velocity d dt of the
oscillating body must be in phase with the excitation force f e . (Note that B is real
positive; we exclude here the special cases of B  0 as being of no practical interest.)
We replace X   iFe (2 B) in Eq. (3.18) and obtain
  2 (m  A)  i ( B  C )  (  gScs  K )  i2 B .
(3.33)
If we separate the real parts from the imaginary parts, we find
 gScs  K

(3.34)
m A
and
B C.
(3.35)
Equation (3.34) is a resonance condition. Compare with the classical case of a simple
mechanical oscillator consisting of a body of mass m hanging from a fixed structure
through a linear spring of stiffness K, as represented in Fig. 3.4. If the body is displaced
from its rest position, it will freely oscillate with frequency
K

.
(3.36)
m
Equations (3.34) and (3.36) are similar to each other, except for the presence, in Eq.
(3.34), of the added mass A and of the buoyance restoring force coefficient  gScs .
8
Equation (3.34) expresses a resonance condition: the incident wave frequency  must
be equal to the frequency of the free oscillations of a mass-spring mechanical system of
mass m  A and spring stiffness  gScs  K .
K
m
of
Fig. 3.4. Simple mass-spring mechanical oscillator.
Condition (3.35) means that the radiation damping must be equal to the PTO
damping for maximum wave energy absorption. We may say that a good wave energy
absorber must also be a good wave radiator.
If the optimal conditions (3.34) and (3.35) are satisfied, the following expression is
obtained from Eq. (3.31) for the maximum time-averaged absorbed power from the
waves (or power output from the PTO)
1
2
Pmax 
Fe .
(3.37)
8B
In wave energy absorption, the concept of efficiency may be misleading and should
be used with care, since the available power is not as well defined as in other energy
conversion systems. Especially in the case of “small” wave energy converters, it may be
more adequate to define an absorption width (or capture width) L as the ratio between
the time-averaged absorbed power P and the energy flux Pwave per unit crest length of
the incident waves
P
L
.
(3.38)
Pwave
This is the width of the two-dimensional wave-train having the same mean power as the
body extracts. If P is in kW and Pwave in kW/m, the absorption width L is expressed in
metres.
3.3.4. Axisymmetric body
An important class of single-oscillating-body wave energy converters have a vertical
axis of symmetry and extract energy essentially through their heave motion. In most
cases their mooring system also allows other modes of oscillation (namely surge and
pitch, see Fig. 3.1) but this will be ignored here.
9

x
O
C
K
Fig. 3.5. Wave energy absorption by a heaving axisymmetric body.
Such a system is represented in Fig. 3.5. As above, the PTO consists of a linear
damper and a linear spring. Since the system is insensitive to the incident wave
direction, the excitation force amplitude   Fe Aw per unit incident wave amplitude
is not a function of the incidence angle. Equation (3.24) becomes simply
 k 2
(3.39)
B
2 g 2 D(kh)
or, in the case of deep water,
 3 2
.
(3.40)
B
2 g 3
Equation (3.37), together with (3.39), gives for the maximum time-averaged absorbed
power
 g 2 Aw2 D(kh)
(3.41)
Pmax 
4 k
or, in terms of absorption width, Lmax  Pmax Pwave . Making use of Eqs (2.58) and
(2.66), we obtain
1 
Lmax  
(3.42)
k 2
(where k is the wavenumber and  is the wavelength), which shows that the maximum
absorption width by an axisymmetric heaving body is equal to  2 and is independent
of the size and shape of the body. This important theoretical result was obtained
independently by Budal and Falnes [3.5], Evans [3.6] and Newman [3.7]. It shows that,
theoretically, a heaving body can absorb more power than the energy flux of an incident
regular wave train along a frontage wider than the width of the body itself. This is a
good reason why the concept of efficiency should be avoided or used carefully in wave
energy absorption.
10
3.4. Time-domain analysis of wave energy absorption by a single
heaving body
If the power take-off system is not linear (the force f PTO is not a linear functional
of the body velocity d d t or of coordinate  ), then the frequency-domain analysis
cannot be employed. In particular, even in the presence of regular incident waves, the
body velocity is not a simple harmonic function of time. In such cases, we have to resort
to the so-called time-domain analysis to model the radiation force.
When a body is forced to move in otherwise calm water, its motion produces a
wave system (radiated waves) that propagate far away. Even if the body ceases to move
after some time, the wave motion persists for a long time (theoretically for ever if
dissipative effects are neglected) and produces an oscillating force of the body wetted
surface that depends on the history of the body motion through the induced radiated
wave field. We are in the presence of a memory effect. This dependence can be
expressed in the following form
t
f (t )  
g (t   ) ( ) d  A()(t ) ,
(3.43)
r

r
where (t ) and (t ) are the body velocity and acceleration, and A() is the value of
the added mass for infinite frequency. In the convolution integral, the velocity is
multiplied by the weighing function g r that accounts for the memory effect and is
expected to tend to zero as its argument increases to infinity. Naturally, the radiation
force f r (t ) at time t can only depend on the body velocity at instants before time t ;
this is why the upper limit of the integral is taken equal to t. The convenience of adding
the term  A()(t ) on the right-hand side of Eq. (3.43) will become apparent below.
To obtain an expression for function g r , we replace, as in the previous section,


f r (t )  Fr ei t   2 A( )  i B( ) X ei t
(3.44)
( )  i X ei ,
(3.45)
and
We obtain
(t )   2 X ei .
 A()  A()  i B()X e 
2
i t
t
g (t   ) ei d
 r
 i X 
.
(3.46)
Changing the integration variable from  to s  t   , we have

i  A( )  A()   B( )   gr ( s) ei sd s .
0
(3.47)
Since the functions A, B and g r are real, we may write

  A( )  A()    gr ( s) sin  s d s ,
0

B( )   gr ( s) cos  s d s .
0
(3.48)
(3.49)
Note that, as    , sin  s and cos  s become rapidly oscillating functions of s.
Since the memory function g r is finite, the integrals in Eqs (3.47) and (3.48) tend to
zero as    , as can be shown by integration by parts. This agrees with Eq. (3.49) as
B( ) vanishes for infinite frequency. On the other hand, the added mass A( ) in
general remains finite at infinite frequency, and this explains the presence of term
11
 A() on the left-hand side of Eq. (3.48) and the reason why the term  A()(t ) was
added to the right-hand side of Eq. (3.43).
We may assume gr (s) to be an even function and, instead of Eq. (3.49), write
1 
gr ( s) ei s d s .
(3.50)



2
This expresses B as the result of a Fourier transform of g r . The inverse Fourier
transform gives
2 
g r ( s)   B( ) cos  s d .
(3.51)
B( ) 

0
Equation (3.51) allows the memory function g r to be computed if the radiation
damping coefficient B is known as a function of frequency.
Replacing, in the governing equation (3.10), the radiation force f r by its expression
(3.43), we obtain
t
m  A()(t )  f (t )  g (t   ) ( ) d   gS  (t )  f .
(3.52)
e

r
cs
PTO
The force f PTO of the power take-off system on the body is supposed to be prescribed
as a function of time t and/or as a function of the body coordinate  and or the body
velocity  , depending on the type of PTO and on the control strategy and algorithm.
Integro-differential equation (3.52) is to be integrated numerically step-by-step in the
time domain from given initial conditions for  and  .
Equation (3.43) for the radiation damping force in the time domain, together with Eq.
(3.50), generalized to the six modes of oscillation, was derived for the motions of a ship
with zero forward speed by Cummins [3.8]. Equation (3.51) is sometimes called
Cummins equation in wave energy absorption by oscillating bodies.
3.5. Wave energy conversion in irregular waves
So far, we considered only sinusoidal or regular waves. Real ocean waves are not
regular: they are irregular and largely random. However, in linear wave theory, they can
be analysed by assuming that they are the superposition of an infinite number of
wavelets with different frequencies and directions. As we saw in section 2.6, the
distribution of the energy of these wavelets when plotted against the frequency and
direction is called the wave spectrum. More precisely, the wave distribution with respect
to the frequency alone, irrespective of the wave direction, is called the frequency
spectrum, whereas the energy distribution as a function of both frequency and direction
is called the directional wave spectrum. Here, we consider only frequency spectra.
We recall that the variance density spectrum is (see Eq. (2.70))
1 1 2
S f ( f )  lim
ai
(3.57)
f 0 f 2
or, in terms of radian frequency  ,
1 1 2
S ( )  lim
ai .
(3.58)
 0  2
In computations, it is convenient to replace the continuum spectrum by a
superposition of a finite number of sinusoidal waves of different amplitudes and
frequencies whose total energy matches the spectral distribution. For that, we divide the
12
frequency range of interest into a set of N small intervals i    i 1
(i  1, 2,..., N  1) of width i  i 1  i and write
1
S ,i i  Aw2 ,i or A ,i  2S ,i i ,
2
where
S ,i  S (ˆi )
(3.59)
(3.60)
and ˆ i  12 (i  i 1) . To simulate the excitation force f e (t ) due to incident irregular
waves characterized by a variance density spectrum S ( ) , we write
N
f e (t )   (ˆ i ) A ,i exp i(ˆ i t   i ) ,
(3.61)
i 1
where ( )  Fe Aw is the excitation force amplitude per unit incident wave
amplitude that is supposed to be known as a function of the frequency  . In Eq. (3.61),
 i is a phase constant taken equal to a random number in the interval (0, 2 ) .
In the time-domain analysis, Eq. (3.61) is written as
N
f e (t )   (ˆ i ) A ,i cos(ˆ i t   i ) .
(3.62)
i 1
It should be noted that Eqs (3.61) and (3.62) provide realizations of the excitation
force that can be used in numerical simulations. Such realizations are not supposed to
reproduce time series of some particular real situation. By choosing different sets of
random phases  i , or by dividing the frequency interval differently, we obtain different
realizations.
In the case of a linear PTO with a linear damper coefficient C, the power, averaged
over a sufficiently long time interval, is
2
1 N 
 2
 d 
P  PPTO     C  i2 X (i ) ,
2 i 1
 dt 
where
X (ˆ i ) 
(ˆ i ) A ,i
 ˆ i2 (m  A(ˆ i ))  iˆ i ( B(ˆ i )  C )  (  gScs  K )
(3.63)
(3.64)
and it was taken into account that
 1
1 t
if i  j
ˆ
ˆ
sin(

t


)
sin(

t


)
d
t

,
2
i
i
j
j

t  t 0
0 if i  j.
lim
Exercise 3.1.
floater
(3.65)
Wave energy absorption by a hemipherical heaving
The heaving hemisphere (Fig. 3.7) is one of the few geometries for which
analytically obtained results are available for the hydrodynamic coefficients of added
mass A and radiation damping B [3.11]. These results are presented in dimensionless
form in Table 3.1 for a sphere of radius a in deep water, where
A( )
B( )
A * (ka) 
,
B
*
(
ka
)

.
(3.66)
2  a3 
2  a3 
3
3
13

a
x
O
C
K
Fig. 3.7. Wave energy absorption by a hemispherical heaving floater.
Table 3.1. Dimensionless coefficients of added A * (ka) mass and radiation damping
B * (ka) versus dimensionless radius ka for heaving hemisphere in deep water (from
[3.11]).
ka
0
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.2
1.4
1.6
1.8
2.0
2.5
3.0
4.0
5.0
6.0
7.0
A * (ka)
0.8310
0.8764
0.8627
0.7938
0.7157
0.6452
0.5861
0.5381
0.4999
0.4698
0.4464
0.4284
0.4047
0.3924
0.3871
0.3864
0.3884
0.3988
0.4111
0.4322
0.4471
0.4574
0.4647
14
B * (ka)
0
0.1036
0.1816
0.2793
0.3254
0.3410
0.3391
0.3271
0.3098
0.2899
0.2691
0.2484
0.2096
0.1756
0.1469
0.1229
0.1031
0.0674
0.0452
0.0219
0.0116
0.0066
0.0040
8.0
9.0
10.0

0.4700
0.4740
0.4771
0.5
0.0026
0.0017
0.0012
0
Table 3.1 shows that, at infinite frequency, the added mass is non-zero (it is exactly
A * ()  1 2) whereas the radiation damping coefficient is zero, as expected. The table
is complemented by the following formulae based on asymptotic expressions supplied
in [3.11] for large and small values of ka
A * (ka)  0.8310  0.75ka ln( ka)  1.266 ka  1.433(ka) 2 for ka  0.1,
A* (ka)  0.5  0.1875(ka) 1  0.44101(ka) 2  0.2608(ka) 3 for ka  8,
B * (ka)  0.75π ka  5.979(ka) 2  5.903(ka)3 for ka  0.2 ,
B * (ka)  13.5(ka) 4  6.75(ka) 5  122.5(ka) 6 for ka  5.
In deep water, it is ka   2a g   *2  (2 T *)2 , where *   a g is a
dimensionless frequency and T *  T g a is a dimensionless wave period.
These numerical values for the added mass and radiation damping coefficient,
together with the equations derived in sections 3.3 to 3.5, may be used to simulate the
performance of a floating hemispherical wave energy converter oscillating in heave.
Results are plotted in Figs 3.8 and 3.9 for a hemispherical heaving floater in deep
water subject to regular waves, with a PTO consisting of a linear damper with
coefficient C, without any spring ( K  0) . In each figure, curves are given for three
values of the dimensionless PTO damping coefficient defined as
C
C*  5 2 1 2 .
(3.67)
a g
1.0
C*  0.5
C*  0.5
0.8
C*  1.0
P
Pmax
0.6
0.4
C*  2.0
0.2
0.0
4
6
8
10
12
T*
Fig. 3.8. Dimensionless plot of time-averaged absorbed power versus wave period for
the floater represented in Fig. 3.7, and for three values of the dimensionless PTO
damping coefficient C * . No spring is present ( K  0) .
15
1.2
C*  0.5
1.0
X
Aw
C*  1.0
0.8
0.6
C*  2.0
0.4
0.2
0.0
4
6
8
10
12
T*
Fig. 3.9. As in Fig. 3.8 for the dimensionless amplitude X Aw of the heaving floater
displacement.
Figure 3.8 shows that close to maximum power is achieved for C*  0.5 (in fact, it is
P  Pmax for C*  0.510) . For larger values of C * , the curves exhibit lower peaks but
become wider. This indicates that PTO damping coefficient that maximizes the energy
absorbed from irregular waves characterized by a given spectrum may be larger than the
optimal damping in regular waves whose frequency is equal of the peak frequency of
the spectrum.
Figure 3.9 shows that the amplitude of the heaving oscillations decreases with
increasing damping coefficient. Naturally it is expected that the damping force will
increase. The PTO power is damping force times floater velocity. Fig. 3.8 shows that
P Pmax may increase or decrease with increasing C * depending on the dimensionless
wave period T * .





The following calculations are suggested to be performed as exercises.
Reproduce the curves plotted in Figs 3.8 and 3.9 by doing your own programming.
Compute the buoy radius a and the PTO damping coefficient C that yield maximum
power from regular waves of period T  9 s. Compute the time-averaged power for
wave amplitude Aw  1 m.
Assume now that the PTO also has a spring of stiffness K that may be positive or
negative. Compute the optimal values for the damping coefficient C and the spring
stiffness K for a buoy of radius 5 m in regular waves of period T  9 s. Explain the
physical meaning of a negative stiffness spring (in conjunction with reactive
control).
Consider “irregular” waves consisting of a superposition of n sinusoidal waves.
Choose n (for example n  5 ), as well as the amplitudes and phases of each
component. Study the performance of the heaving buoy in irregular waves.
Study the performance of the heaving buoy in irregular waves characterized by the
spectrum of Eq. (3.58). Choose the number of components ( N  200 is a typical
value).
16
Exercise 3.2. Heaving floater rigidly attached to a deeply submerged
body
Wave energy converters whose horizontal dimension is small compared with that the
wavelength are sometimes named point absorbers. In most cases, the resonance
frequency of floating point absorbers is significantly smaller than the typical frequency
of ocean waves. For example, in the case of a heaving hemisphere with a linear PTO
damper, Fig. 3.8 shows that resonance occurs for T *  T g a  6.1 . If the radius is
a  7 m, then it is T  5.16 s, which is significantly less than the typical wave period of
energetic sea states. An alternative to increasing the size of the buoy, is to rigidly
connect it to a submerged body in order to increase the inertia of the pair. If the distance
from the body to the free surface is large enough (say not less than about 30 m) then the
disturbance due the body motions upon the wave field around the floater is small. This
is because the radiated surface-wave field due to the body motion as well as the
excitation force on the body vanish with increasing submergence. This kind of geometry
was adopted in one of the bodies of the two-body device WaveBob, as shown in Fig.
1.26.
This situation is represented in Fig. 3.10. The governing equation in the frequency
domain is Eq. (3.18) with m  A( ) replaced by m  A( )  m1  A1 . Here m1 is the
mass of the submerged body and A1 its added mass. Note that, for a deeply submerged
body, the added mass depends only on body geometry, and is independent of the
frequency  of its oscillations provided that the submergence is not smaller than about
half the wavelength of surface waves of frequency  .

a
x
m1
C
Fig. 3.10. Wave energy absorption by a heaving hemispherical floater attached to a
deeply submerged body.
17
As an exercise, consider a hemispherical floater of radius a  7 m with a linear PTO
damper (Fig. 3.10), and determine the optimal value of m1  A1 (mass plus added mass
of the deeply submerged body) and of PTO damping coefficient C for regular waves of
period T  8 s.
3.6. Wave energy absorption by two-body oscillating systems
3.6.1. Governing equations
The concept of the point absorber for wave energy utilization was developed in the
late 1970s and early 1980s, mostly in Scandinavia. This is in general a wave energy
converter of oscillating body type whose horizontal dimensions are small compared to
the representative wave length. In its simplest version, the body reacts against the
bottom. In deep water (say 40 m or more), this may raise difficulties due to the distance
between the floating body and the sea bottom, also possibly to tidal oscillations of the
surface level. Multi-body systems may then be used instead, in which the energy is
converted from the relative motion between two bodies oscillating differently.
Sometimes the relevant relative motion results from heaving oscillations. This is the
case of several devices like the the Wavebob, the PowerBuoy and the AquaBuoy (see
Chapter 1).
PTO
body 1
body 2
Fig. 3.11. Two-body heaving wave energy converter.
Figure 3.11 represents a two-body wave energy converter in which the oscillations
are essentially heaving. The PTO converts the energy associated to the relative motions
and the forces between the two bodies. The coupling between bodies 1 and 2 is due
firstly to the PTO forces and secondly to the forces associated to the diffracted and
radiated wave fields. It is obvious that the excitation force on one of the bodies is
affected by the presence of the other body. Besides, in the absence of incident waves,
the radiated wave field induced by the motion of one of the bodies produces a radiation
force on the moving body and also a force on the other body.
18
We denote by 1 and  2 the vertical displacements of bodies 1 and 2 from their
undisturbed positions. The governing equations can be written as (see Eqs (3.9) and
(3.10))
m1
d 21
dt 2
 f e,1  f r ,11  f r ,12  g Scs,11  f PTO ,
(3.68)
d 2 2
 f e,2  f r ,22  f r ,21  g Scs,2 2  f PTO.
(3.69)
dt 2
Here, mi (i  1, 2) is the mass of body i, f e,i is the excitation force on body i, Scs,i is
m2
the cross-sectional area of body i defined by the undisturbed free-surface plane, f r ,,ii is
the radiation force on body i due to its own motion, and f r ,ij is the radiation force on
body i due to the motion of body j.
3.6.2. Linear system. Frequency domain analysis
We assume now that the power take-off system is linear and consists of a linear
spring of stiffness K and a linear damper with damping coefficient C, connected in
parallel, as we considered in section 3.3 and is represented in Fig. 3.3. The PTO force
may be written as
d(   )
f PTO  C 1 2  K (1   2 ).
(3.70)
dt
We consider an incident wave of frequency  and amplitude Aw , propagating in the
positive x-direction (from left to right) on deep water or on water of arbitrary but
uniform depth. Since the system is linear and the input (incident wave) is represented by
a harmonic function of time, the bodies’ displacements and the forces upon the bodies
are also harmonic functions of time. We may write
i (t )  X i ei t ,
f e,i (t )  Fe,i ei t ,
f r ,ij (t )  Fr ,ij ei t
(i, j  1, 2),
(3.71)
where X i is the complex amplitude of the displacement  i of body i, and Fe,i and Fr ,ij
are the amplitudes of the excitation forces and radiation forces, respectively. These
amplitudes are in general complex.
As for the single heaving body, we decompose the radiation force coefficient Frij as
Fr ,ij  ( 2 Aij  i Bij ) X j
(i, j  1, 2),
(3.72)
where Aij and Bij are real coefficients of added mass and radiation damping,
respectively. They depend on wave frequency  and on the geometry of the two-body
system. The same commercial software based on the finite-element method (WAMIT,
ANSYS/Aqwa, Aquaplus) can also be used to compute Aij , Bij and Fe,i Aw (and more
generally for any number of bodies and degrees of freedom). As was the case of B in
section 3.3, we may also conclude that the radiation damping coefficients Bii cannot be
negative; the same cannot be said for Bij if i  j . It can be proved (see [3.2]) that the
cross coefficients are equal:
(3.73)
A12  A21, B12  B21.
In the frequency domain, Eqs (3.68) and (3.69) become
19
  (m  A )  i(B  C)  ( g S  K )X
   A  i ( B  C )  K X  F ,
  (m  A )  i(B  C)  ( g S  K )X
   A  i ( B  C )  K X  F .
2
1
11
11
cs1
1
2
12
12
2
(3.74)
e,1
2
2
22
22
cs 2
2
2
12
12
1
(3.75)
e, 2
Equations (3.74) and (3.75) are a system of linear algebraic equations in the
unknowns X 1 and X 2 that can be easily solved. The instantaneous power available to
the PTO is
(3.76)
P
 C (   ) 2  K (   )(   ) .
PTO
1
2
1
2
1
2
In time average, we have, for the absorbed power,
2
P  PPTO  12 C 2 X1  X 2 .
(3.77)
As in subsection 3.3.2, we consider the case in which the direction of the incident
wave propagation makes an angle  (     ) with the x-axis, and denote by
i ( )  Fe,i (  ) Aw the corresponding value of the excitation force amplitude per unit
incident wave amplitude. It may be shown [3.2] that, for fixed  , the following
relationship (Haskind relation) exists between Bii and i (  )

k
i ( )2 d ,
Bii 
(3.78)

2


4  g D(kh)
where D(kh) is given by any of the expressions (2.67). In the deep water limit, it is
kh   , D(kh)  1 and
3 
(3.79)
 ( )2 d .
3  i
4  g
Equations (3.78) and (3.79) are similar to Eqs (3.24) and (3.25) that apply to a single
heaving body. If the system has a vertical axis of symmetry, i is independent of 
Bii 
and Eqs (3.78) and (3.79) become more simply
 ki
Bii 
,
2 g 2 D(kh)
and
 3i
Bii 
(deep water).
2 g 3
(3.80)
(3.81)
In the case of an axisymmetric device, equations (3.80) and (3.81) allow the modulus
of excitation force amplitudes Fe,1 and Fe,2 to be computed from the radiation damping
coefficients B11 and B22 . However, they do not yield their relative phases, which may
be essential in the analysis of a two-body system with two oscillation modes.
Important theoretical results in the frequency domain can be found in [3.12] for twobody heaving wave energy converters.
3.6.3. Time domain analysis
The time domain analysis, required if the PTO is not linear, can be performed as for
a single heaving body. Instead of Eqs (3.43) and (3.51), we write
20
t
g (t
 r ,ij
f r ,ij (t )  
  )  j ( ) d  Aij ()j (t )
(3.82)
and
g r ,ij ( s)  g r , ji ( s) 
2


0
Bij ( ) cos  t d .
(3.83)
Equations (3.68) and (3.69) become
(m1  A11())

d 21
dt
2
t
 f e,1  
t
g (t
 r ,11
g (t
 r ,12
  ) 1 ( ) d  g Scs,11
  ) 2 ( ) d  A12 ()2 (t )  f P TO ,
t
d 2 2
(m2  A22 ()) 2  f e,2   g r ,22 (t   ) 2 ( ) d  g Scs,2 2

dt

t
g (t
  r ,12
(3.84)
(3.85)
  ) 1 ( ) d  A12 ()1 (t )  f P TO.
Exercise 3.3. Heaving two-body axisymmetric wave energy converter
This exercise concerns a heaving axisymmetric two-body device with a linear PTO
absorbing energy from regular waves. Figure 3.12 represents a the system consisting of
two axisymmetric co-axial bodies 1 and 2. This is a simplified representation of some
wave energy converters under development, namely Wavebob and PowerBuoy. Body 1
is a cylindrical floater with a conical bottom, whereas body 2 is a long cylinder with a
flat bottom. The gap between bodies 1 and 2 is very small, but the friction between
them is neglected. The system is assumed to move only in heave. The PTO is driven by
the forces between bodies 1 and 2 and their relative translational motion.
PTO
2a
x
c
body 1
2b
body 2
d
Fig. 3.12. Heaving two-body axisymmetric wave energy converter.
21
To simplify the exercise, the draught d of body 2 is assumed large enough for the
excitation and radiation forces on its flat bottom to be negligible, i.e. Fe,2  0 , B22  0
and A12  0 . However, the added mass A22 is accounted for. We may also assume that
the surface waves radiated by body 2 are negligible and so take B12  0 . Note that we
are interested only on the vertical components of the forces, and that the excitation and
radiation forces on body 1 result from water pressure on its conical wetted surface.
Obviously, such forces are independent of the draught d and of the motion of body 2,
provided that d is large enough.
The PTO consists of a linear damper with damping coefficient C. We consider the
case when a  c , b  0.4a. and the semi-angle of the conical bottom (angle between the
generatrices and the axis) is equal to 60 . The volume of the submerged part of body 1
in calm water is 3.031a3. Dimensionless values of the added mass A11 and radiation
damping coefficient B11 of body 1 are defined as
A
B11
*
*
A11
 113 ,
B11

(3.86)
 a
 a 3
and are plotted versus dimensionless wave period T *  T g a  2 1 g a in
Fig. 3.13 (tables of numerical values are available on request)
*
A11
*
B11
T*
Fig. 3.13. Dimensionless plot of added mass coefficient and radiation damping
coefficient for body 1 represented in Fig. 3.12.
The added mass of a semi-infinite circular cylinder of radius b moving along its axis
in unbounded water of density  has been computed and is equal to 0.6897   b3 .
Assuming the depth of submergence d to be large, the added mass A22 of body 2 may
be taken as independent of  and equal to that value.
The following items are suggested to be performed as exercises.
 Write the governing equations in the frequency domain.
 Compute the mass m2 of body 2 as a function of submergence d.
 For given dimensionless wave period T * , find the optimal values of the ratio d a
and of the dimensionless PTO damping coefficient
C
C*  5 2 1 2 .
a g
22
 Discuss the advantages and limitations of a wave energy converter based on this
concept.
3.6. Wave energy absorption by oscillating systems with several
degrees of freedom
In the preceding sections, we analyzed one body and two-body wave energy
converters oscillating in heave. These results can be generalized to include the other five
modes of oscillation and can be extended to any number of bodies, with different types
of linear and non-linear power take-off systems. This kind of general analysis, based on
linear water wave theory, can be found in detail in the book by Falnes [3.2].
We note that, of the six degrees of freedom of a body, three are rotations: pitch, roll
and yaw (see Fig. 3.1). For these rotations, instead of forces we have moments, and
instead of added mass, we have added moment of inertia. There can be interference
between modes of motion through the diffracted and radiated wave fields and also
possibly through the PTO or PTOs and moorings.
In the case of a floating body with a vertical plane of symmetry parallel the direction
of the incident waves, the excited modes of motion are heave, surge and pitch. For a
body with a vertical axis of symmetry, heaving oscillations do not induce surge and
pitch oscillations
3.7. Time-domain analysis of a heaving buoy with hydraulic PTO2
3.7.1. Introduction
The energy of sea waves can be absorbed by wave energy converters in a variety of
manners, but in every case the transferred power is highly fluctuating in several timescales, especially the wave-to-wave or the wave group time-scales. In most devices
developed or considered so far, the final product is electrical energy to be supplied to a
grid. So, unless some energy storage system is available, the fluctuations in absorbed
wave power will appear unsmoothed in the supplied electrical power, which severely
impairs the energy quality and value from the viewpoint of the grid. Besides, that would
require the peak power capacity of the electric generator and power electronics to
greatly exceed the time-averaged delivered power.
In practice, three methods of energy storage have been adopted in wave energy
conversion. An effective way is storage as potential energy in a water reservoir, which
is achieved in some overtopping devices, like the Wave Dragon and the SSG. In the
oscillating water column type of device, the size and rotational speed of the air turbine
rotor make it possible to store a substantial amount of energy as kinetic energy
(flywheel effect).
In a large class of devices, the oscillating (rectilinear or angular) motion of a floating
body (or the relative motion between two moving bodies) is converted into the flow of a
liquid (water or oil) at high pressure by means of a system of hydraulic rams (or
equivalent devices). At the other end of the hydraulic circuit there is a hydraulic motor
(or a high-head water turbine) that drives an electric generator. The highly fluctuating
hydraulic power produced by the reciprocating piston (or pistons) may the smoothed by
2
This section is largely based on [3.13] and [3.14].
23
the use of a gas accumulator system, which allows a more regular production of
electrical energy. Naturally the smoothing effect increases with the accumulator volume
and working pressure. This kind of power take-off system is employed e.g. in the
Pelamis, the Wavebob and the PowerBuoy.
Here we analyse the performance of a floating oscillating body wave energy
converter with one degree of freedom (heave). The buoy motion drives a two-way
hydraulic ram that feeds high pressure oil to a hydraulic motor (or water to a high-head
hydraulic turbine). A gas accumulator system is placed in the circuit to produce a
smoothing effect. Such a wave energy converter is highly non-linear, which requires a
time-domain model consisting of a set of coupled equations: (i) an integral-differential
equation (with a convolution integral representing the memory effect) that accounts for
the hydrodynamics of wave energy absorption; (ii) an ordinary differential equation that
models the time-varying gas volume and pressure, the dependence of flow rate
(supplied to the motor or turbine) on pressure head, the non-return valve system, and the
pressure losses in the hydraulic circuit (viscosity effects). In the case of several degrees
of freedom (not considered here), additional (differential and/or integral-differential)
equations appear. Standard methods are employed to numerically integrate the
differential equations, with appropriate initial conditions.
Random irregular waves are assumed (each sea state is characterized by its
significant wave height H s and energy period Te , and a discretized Pierson-Moskowitz
spectrum). A simple geometry (a hemisphere in deep water oscillating in heave) is
adopted for the buoy.
3.7.2. Governing equations
We consider the simple case of a body of mass m with a single degree of freedom
oscillating in heave (coordinate  , with   0 in the absence of waves). The PTO
consists of a hydraulic comprising a ram or hydraulic cylinder, a manifold (valve
system), high- and low-pressure gas accumulators and a hydraulic motor (Fig. 3.14).
The hydraulic motor operates between the two accumulators.
HP gas
LP gas
accumulator accumulator
Buoy
Motor
E
D
Valve
A
A
Cylinder
B
B
Fig. 3.14. Heaving buoy with hydraulic PTO.
The equation governing the motion of the buoy is Eq. (3.52), derived in section 3.4,
t
m  A()(t )  f (t )  g (t   ) ( ) d   gS  (t )  f ,
(3.52)
e

r
cs
24
PTO
where the memory function is given by Eq. (3.51)
2 
g r ( s)   B( ) cos  s d .
(3.51)
 0
Numerical values for the added mass A() and radiation damping coefficient B( ) for
a heaving hemisphere are given in Exercise 3.1. The memory effect decays rapidly with
time, and may be neglected after a few tens of seconds (the infinite interval of
integration in the convolution integral of Eq. (3.52) may be replaced by a finite one).
For identical reasons, a finite interval of integration is kept in Eq. (3.51) (an upper limit
of about 3-5 rad/s is probably enough in practice).
In the case of an axisymmetric heaving body in deep water (as is the case here)
subject to incident regular waves of frequency  and amplitude Aw , the modulus
Fe ( )  ( ) Aw of the complex amplitude of the excitation force is related to the
radiation damping coefficient B( ) by the Haskind relation (3.40)
 3 ( )2
(3.40)
B( ) 
2 g 3
which may be used to compute ( ) from the given numerical values of B( ) .
We assume the wave spectral distribution to be given, for example the PiersonMoskowitz spectrum S ( ) of Eq. (3.58)


S ( ) 262.6 H s2 Te4 5 exp  1052 (Te  ) 4 .
1 2
Aw,i or A ,i  2S ,i i ,
2
The excitation force f e (t ) may be calculated as in sub-section 3.5.2:
S ,i i 
(3.58)
(3.59)
N
f e (t )   (ˆ i ) A ,i exp i(ˆ i t   i ) ,
(3.61)
i 1
where A ,i  2S ,i i and S ,i  S (ˆ i ) .
In the numerical simulations, the spectrum was discretized into 225 equally spaced
(  i  0.01rad/s)
sinusoidal
harmonics
in
the
range
0.1 6    0.1 6  2.24 rad/s . (The irrational number 6 ensures the non-periodicity
in the time-series of f e (t ) .) The phases at t  0 were made equal to random numbers in
the interval (0,2 ) . The integral-differential equation (3.52) was numerically integrated
in the time domain with a time step size of 0.1s .
All the numerical results presented in this section are for a hemispherical floater of
radius a  5 m. The hydrodynamic coefficients A() and B( ) were obtained from
[3.11] (see Exercise 3.1).
3.7.3. The power take-off mechanism
In most wave energy converters using hydraulic rams as mechanical power take-off
system, the displacement of the piston inside the corresponding cylinder is driven by the
relative motion between two oscillating bodies. In this case, there is only one oscillating
body, and the cylinder (or alternatively the piston) is fixed (with respect to the sea
bottom or to a shoreline structure).
25
The hydraulic circuit includes a high-pressure (HP) gas accumulator, a low-pressure
(LP) gas accumulator and a hydraulic machine (Fig. 3.14). The machine can be either a
hydraulic motor (if the working fluid is oil) or a high-head water turbine. A rectifying
valve system prevents liquid from leaving the HP accumulator at E and from entering
the LP accumulator at D. In this way, when the piston is moving downwards, the liquid
in pumped from the cylinder into the HP accumulator through the duct BE and sucked
from the LP accumulator into the cylinder through DA. During the upward motion,
the circuit is AE and DB. The hydraulic machine is driven by the flow resulting
from the pressure difference between the HP and LP accumulators.
Let p  p  g where p and  are pressure and vertical coordinate in the liquid
circuit (   0 at average sea surface level). We denote by pa , pb , p1 and p2 the
values of p in the upper and lower parts of the cylinder, and inside the HP and LP
accumulators, respectively.
First we consider an interval of time when the piston is moving upwards (flow
directions AE and DB), which implies that pa  p1 and pb  p2 . The volume flow
rate is q  Sc d dt , where S c is the cylinder cross-sectional area, and the coordinate
 defining the piston position (and also the floater position) increases upwards. (Here
we neglect the cross-sectional area of the piston rod. It should be noted however that, in
very high pressure oil hydraulics, the rod-to-piston diameter ratio is usually not small
and can be as high as about 0.5; in such cases, some of the equations presented here
should be modified to take the rod cross-sectional area into account.) Assuming onedimensional flow, we may write
dq
p1  p2  pa  pb  ku q 2  I
.
(3.87)
dt
Here, ku is a coefficient of pressure loss due to friction along the circuit, and I is a
coefficient that takes into account the inertia of the fluid. Likewise, if the piston is
moving downwards (flow directions DA and BE), it is
dq
p1  p2  pb  pa  kd q 2  I
.
(3.87)
dt
We assume that ku  kd  k . Then, regardless of the direction of the piston motion, we
may write
dq
p1  p2  p  kq2  I
,
(3.88)
dt
where p  pa  pb . Whenever p  p1  p2 , it is q  0 , i.e. the valve prevents the
piston from moving.
The hydraulic machine will be driven by the pressure difference p1  p2 . In the case
of an impulse hydraulic turbine (Pelton turbine), the flow rate is independent of the
rotational speed, and may be written as
12
 p  p2 

qm   1
 Kt 
,
(3.89)
where Kt  An1 (2n  w ) 1 2 . Here  w is density of water in the circuit, An is the
effective cross-sectional area of the turbine nozzle (or nozzles) (which may be
controlled) and  n is the nozzle efficiency (that accounts for losses in the nozzle or
nozzles and also in the connecting duct from the accumulator).
26
In the case of a hydraulic motor, the flow rate is approximately proportional to the
rotational speed  , and we may write qm  m  , where m is a constant
characterizing the machine geometry. (There are variable-geometry hydraulic motors
that allow the rotational speed and the flow rate to be controlled separately.)
We denote by m1 and m2 the masses of gas inside the HP and LP accumulators,
respectively, which are supposed to remain unchanged during operation. Assuming the
duct and accumulator walls to be rigid and the liquid incompressible, the total volume
of gas remains constant, i.e. m1v1 (t )  m2v2 (t )  V0  constant ( vi , i  1,2, is specific
volume of gas). We may also write
dv (t )
q(t )  qm (t )  m1 1 .
(3.90)
dt
The specific entropy s1 of the gas inside the HP accumulator will change due
essentially to heat transfer. This may be connected to changes in sea water temperature
and surrounding air temperature, and also to changes in the power dissipated (viscous
losses and electrical losses) inside the converter. Such changes are likely to be
significant over time intervals not less than several hours, and so it is reasonable to
consider that the gas compression/expansion process inside the accumulator is
approximately isentropic ( s1, s2 are constant) during a sea state (this means that,
although the changes in gas temperature may be significant during the
compression/expansion cycle, the corresponding changes in entropy may be neglected).
For an isentropic process of a perfect gas, it is  i (t )  vi (t )  i ( i  1, 2 ), where  i is
gas pressure,  i is constant for fixed entropy si , and   c p cv is the specific-heat
ratio for the gas. We assume that z  0 at the liquid free-surface inside the HP and LP
accumulators, and so it is  i  pi ( i  1, 2 ). From Eq. (13), it follows that

V  m1v1(t ) 
dq(t )
(3.91)
p(t )  1v1(t )  2  0
 Kq(t )2  C
.

m2
dt


We note that the force Sc  pa (t )  pb (t ) required to pump fluid into the HP
accumulator is to be overcome by the action of the buoy upon the piston.

3.7.4. Floating converter with gas accumulator
We consider again the buoy oscillating in heave and driving a hydraulic cylinder or
ram that pumps high pressure liquid (oil or water) into a hydraulic circuit (Fig. 3.14).
The rectifying valve is controlled in such a way that the liquid is pumped from the
cylinder into the HP accumulator and sucked from the LP accumulator into the opposite
side of the cylinder. The turbine or the rotary hydraulic motor is driven by the flow
resulting from the pressure difference between the HP accumulator and the LP
accumulator. The time variation of the gas pressure difference p1 (t )  p2 (t ) between
the HP and LP accumulators results from (i) the action of the buoy upon the piston, and
(ii) the flow of liquid through the turbine or hydraulic motor. For simplicity, we neglect
the inertia and the pressure losses in the hydraulic circuit, i.e. set I  0, k  0 in Eq.
(3.88), and so pa  pb  p1  p2 whenever the piston is moving. (The inertia of the
liquid in the hydraulic circuit could be modelled by a mass to be added to the mass of
the buoy, m, in Eq. (3.52).)
27
While the body is moving, the governing equation is (3.52), with f PTO   sign( x ) ,
where   Sc ( p1  p2 ) and S c is the cylinder cross-sectional area. At some time, the
time-varying body velocity will be zero. From then on, the body will remain stationary
unless, or until, the hydrodynamic force on the body
f e (t )   gScs  (t ) 
t
 g r (t   )( ) d
(3.92)

overcomes the resisting force   Sc ( p1  p2 ) and fluid is again pumped into the HP
accumulator (this has been named Coulomb damping force).
The instantaneous power absorbed by the converter is
(3.93)
P(t )   (t )
and its time-average in t0  t  t f is
tf
P  t 1  P(t ) dt .
(3.94)
t0
The value of P naturally depends on the magnitude of the time interval t  t f  t0 .
3.7.5. Control
It is important to control the device in order to maximize the produced energy. This
should take into account the sea state, characterized by H s and Te . Since the system is
assumed linear from the hydrodynamic point of view, then, for fixed Te , the values of
P H s2 and q H s will depend only on the ratio  H s (we assume the force  to be
approximately constant over the sea state under consideration, and, as before, q  S 
c
is the flow rate pumped by the piston).
The relationship P H s2  f P ( H s , Te ) is represented in Fig. 3.15 (time-averages
over 15 min), for a  5 m and Te  5 , 7, 9, 11 and 13s. It may be seen that the optimal
value of ( H s )opt (i.e. that maximizes P H s2 ) varies with Te .
12
10
Te=5s
Te=7s
Te=9s
Te=11s
Te=13s
G=G1
G=G2
G=G3
8
6
4
2
0
0
50
100
150
 H s (kW m)
200
250
Fig. 3.15. Converter performance with simple Coulomb-type damping. Performance
curves (for Te  5  13s ) and control curves (parabolas).
28
It should be noted that q  qm ( qm  flow rate through the hydraulic machine) over a
sufficiently large time span, and also that P   x  q Sc  qm Sc . So, for fixed
Te , qm (Sc H s2 ) may also be regarded as a function of  H s and the same obviously
applies to
(3.95)
qm Sc   ( H s )2 f P ( H s , Te ) .
For each value of Te , Fig. 3.15 and Eq. (3.95) yield the optimal value for the ratio
(qm Sc )opt . This may provide a control algorithm for the flow rate qm versus the
pressure difference p1  p2   Sc . For practical reasons, it may be convenient to have
a control law independent of the wave period Te , and so it may be reasonable to adopt
instead a single value (i.e. independent of Te ) for qm Sc   Sc2G (G  constant) . This
appears in Fig. 3.15 as a regulation curve
2
P
 qm   
 G,
(3.96)


2
H s Sc H s2  H s 
which is a parabola. Figure 3.15 shows four such curves for G  Gi (i  1 to 4), with
G1  0.4  106 , G2  0.6  106 , G3  1.0  106 and G4  2.0  106 s/kg . The value of
P H s2 is given by Fig. 3.15 as the intersection between the appropriate Te -curve and
the parabolic regulation curve defined by the chosen value for G. The value G2 may be
regarded as an acceptable compromise, especially in the range 7  Te  13 s (see Fig.
3.15).
In order to establish a control strategy based on this algorithm (which was devised
assuming constant force  over the sea state under consideration), we assume now that
the gas accumulator is large enough so that the variations in p1  p2 may be neglected
over a few wave periods, and define individual “sea states” of such duration. Then we
adopt
qm (t )Sc (t )  qm (t ) p1(t )  p2 (t ) Sc2G  constant
(3.97)
as an instantaneous control algorithm. We note that Eq. (3.97) is a linear relationship
between qm and p1  p2 , differently from the square-root relation (3.89). This means
that, if a Pelton water turbine is employed, the exit nozzle-area has to be controlled.
This control algorithm was numerically tested for a  5 m, Sc  0.01767 m2
(cylinder
of
0.15 m
inside
diameter),
G  G2  0.6  106 s/kg ,
m1  150 kg ,
m2  30 kg, V0  5.17 m3 (total volume of gas). For each sea state, the values of 1
and  2 (that depend only on the specific entropy) were chosen such that the timeaveraged temperatures of the gas in the HP and LP accumulators realistically remained
close to environmental temperature (  300 K ). Figure 3.16 shows results for a sea state
characterized by H s  1.5 m and Te  11 s . The values of P(t ) H s2  ( p1  p2 ) q H s2
and of Pm (t ) H s2  ( p1  p2 )qt H s2  Sc2G( p1  p2 ) H s2 ( Pm is power available to the
hydraulic machine), averaged over a time span of 2 hours, were found to be
P H s2  10.21 kW/m2 and Pm H s2  10.25 kW/m2. These values closely agree with
those given by Fig. 3.15. The computed standard deviation of the power available to the
29
hydraulic machine is  Pm  0.0811Pm , i.e. 8.11% of its time-averaged value. The timeaveraged values of the HP and LP accumulator gas pressures are p1  127.4 bar and
p2  15.6 bar.
The corresponding values, computed for Te  11 s, H s  4 m and the same 2-hour
period of time, are P H s2  10.17 kW/m2 and Pm H s2  10.20 kW/m2, practically
showing no change with H s . However, the fluctuations in Pm H s2 are now much
larger:  Pm  0.293Pm , which indicates that the smoothing effect of the accumulator
decreases markedly in more energetic sea states. The accumulator pressures are now
p1  301.0 bar, p2  11.1 bar.
Fig. 3.16. Performance results for H s  1.5 m, with controlled qm . In the bottom graph,
the chain curve represents the power Pm available to the hydraulic machine.
3.7.6. Phase control by latching
Phase-control by latching has been proposed by Budal and Falnes [3.15] to
enhance the wave energy absorption by oscillating bodies (namely the so-called
point absorbers) whose natural frequency is above the range of frequencies within
which most of the incident wave energy flux is concentrated. Later, this has been
confirmed experimentally. Phase control by latching was the object of other
theoretical and experimental investigations (see e.g. [3.16]). Although phase control
30
by latching has been shown to be potentially capable of substantially increasing the
amount of absorbed energy, the practical implementation in real irregular waves of
optimum phase control has met with theoretical and practical difficulties that have
not been satisfactorily overcome. Sub-optimal control methods have been devised
and proposed by several research teams to circumvent such difficulties.
The use of a hydraulic power take-off (PTO) system as described above provides
a natural way of achieving latching: the body remains stationary for as long as the
hydrodynamic forces on its wetted surface are unable to overcome the resisting
force (gas pressure difference p times cross-sectional area S c of the ram)
introduced by the hydraulic PTO system.
Phase control by latching is implemented by adequately delaying the release of
the body in order to approximately bring into phase the body velocity and the
diffraction (or excitation) force on the body, and in this way get closer to the wellknown optimal condition derived from frequency-domain analysis for an oscillating
body in regular waves, with linear PTO damping (see Eq. (3.32)). The proposed
control algorithm is simple and easy to implement, and includes (i) a proportionality
relationship qm  C1p between the fluid flow rate q m through the hydraulic
motor (or water turbine) and the accumulator gas pressure difference p , and (ii) a
proportionality relationship F  C2 p between the release force F and p (which
regulates the release delay).
When the body is moving, its velocity will, at some time, come to zero, as a result of
the hydrodynamic forces on its wetted surface and the PTO forces. The body will then
remain fixed until the hydrodynamic force f h exceeds R  R(Sc p) , where R  1 . It
is to be noted: (i) that the force that has to be overcome (if the body is to restart moving)
is now larger (by a factor R) as compared with the simple Coulomb damping (i.e.
compared with Sc p ); (ii) that the acceleration of the floater (unlike in the case of
R  1 ) is discontinuous when the body is released. There is now a new parameter, R, to
be optimized, jointly with parameter G.
Numerical simulations (30 min each) were carried out, based on this procedure and
algorithm, for a hemispheric floater of radius a  5 m, in deep water, in regular and
irregular waves. Piston area was Sc  0.0314 m2. The masses of gas (nitrogen) in the
HP and LP accumulators were m1  100 kg and m2  20 kg. In each simulation, the
values of gas entropies s1 and s 2 were taken such that the time-averaged gastemperatures in the HP and LP accumulators remained close to environmental
temperature (  300 K).
Regular waves.
Results of simulations in regular waves (wave amplitude Aw  0.667 m and period
T  9 s) are shown in Figs. 3.17 to 3.21. In Fig. 3.17, the solid line shows the
numerically optimized values (that maximize P ) of control parameter G for several
values of the latching control parameter R (note that R  1 means simple Coulomb
damping). In the same figure, the dashed line represents the amplitude of oscillation
xmax . Figures 3.18 and 3.19 represent, for the same situations, the time-averaged
absorbed power P and the time-averaged gas pressure (in the HP accumulator) p1 ,
respectively, versus R. It may be seen that, by increasing R above unity and (for each R)
suitably optimizing G, a substantial increase (by a factor up to about 3.8) in the time31
averaged absorbed power P can be achieved. The maximum power attained in this
way, about 206 kW for R  16 , should be compared with the theoretical maximum
power 14 g 3 Aw2  3  315 kW absorbed by an axisymmetric body with a linear PTO
damper oscillating in heave. It is to be noted that this increase in absorbed power results
mostly from larger floater oscillations  max (and hence greater liquid flow through the
hydraulic motor) rather than from greater pressure levels in the HP hydraulic circuit.
Figures 3.20 and 3.21 represent the time variation of the excitation force fe (t ) , the
floater velocity d t and its displacement  (t ) for two situations optimized with
respect to G (same incident wave time series, for easier comparison): R  1 ,
G  0.86 10 6 s/kg (simple Coulomb damping, Fig. 3.20) and R  16 , G  7.7 106
s/kg (latching control, Fig. 3.21). It is to be noted that, in Fig. 3.21 (but not in Fig. 3.20),
the velocity and the diffraction force are (very approximately) in phase with each other
(in agreement with optimal condition (3.32) for linear PTO).
G 106 (s kg)
 max Aw
Fig. 3.17. Regular waves, Aw  0.667 m , T  9 s : optimized control parameter G 106
(solid line) and dimensionless oscillation-amplitude,  max Aw (dashed line), versus
latching control parameter R.
Fig. 3.18. As in Fig. 3.17: time-averaged absorbed power P (kW) versus R (G
optimized for each R).
32
Fig. 3.19. As in Fig. 3.17: time-averaged gas pressure in HP accumulator versus R (G
optimized for each R).
10  f e (MN)
d dt (m/s)
 (m)
Fig. 3.20. Performance of a hemispherical floater in regular waves for R  1,
G  0.86  106 s/kg . Above, d dt : solid line, fe (t ) : broken line. Below,  (t ).
Absorbed power: P  55.0 kW.
33
Fig. 3.21. As in Fig. 3.20, for R  16, G  7.7  106 s kg . P  206.1 kW.
Fig. 3.22. Hemispherical floater in irregular waves, Te  7 s. Plot of P H s2 versus
control parameter G, for several values of latching control parameter R.
34
Fig. 3.23. As in Fig. 3.22, for Te  9 s.
Fig. 3.24. As in Fig. 3.22, for Te  11 s.
Irregular waves
Optimal phase control in random irregular waves is known to require the prediction
of the incoming waves (theoretically over the infinite future, in practice over a few tens
of seconds, see [3.17]). In addition to this difficulty, the theoretical determination of the
wave-to-wave optimal latching period requires relatively heavy computation, which
makes it inappropriate for implementation in real time.
Therefore, it is particularly interesting to investigate whether the simple control
strategy outlined above, and tested above in regular waves, can be applied successfully
to irregular waves. One ought to bear in mind that this should be regarded, at best, as a
sub-optimal strategy, and that the achievable results should not be expected to be close
to the theoretical maximum.
Numerical simulations, identical to those presented above for regular waves, were
performed for irregular waves as modelled by a Pierson-Moskowitz spectrum and
H s  2 m, Te  7, 9 and 11s. The results, in terms of time-averaged absorbed power P
divided by H s2 , are presented in Figs. 3.22 to 3.24.
35
d
dt  H s
10  f e (MN)
 Hs
Fig. 3.25. Performance of a hemispherical floater in irregular waves for Te  9s , R  1,
G  0.7  106 s/kg . Above, d dt : solid line,
fe (t ) : broken line. Below,  (t ).
Absorbed power: P H s2  10.3 kW/m 2 .
The figures show that a large increase (by a factor about 2.3-2.8) in absorbed power
(as compared with simple Coulomb damping, R  1 ) can be achieved by suitably
combining the values of the control parameters R ( R  1) and G. The largest absorbed
power occurs for R equal to about 16 and a value of G that depends on R and Te .
Curves for the excitation force f (t ) , and the floater velocity (t ) and displacement
e
 (t ) are given in Figs. 3.25 and 3.26, for Te  9 s and control parameter pairs
( R  1, G  0.7  106 s/kg) , ( R  16, G  4.2 106 s/kg) .
It is not surprising that those large values of absorbed power occur for relatively
large amplitudes of the floater oscillations, that typically attain nearly twice the value of
the significant wave height H s , as shown in Fig. 3.26.
Since the whole analysis is based on linear hydrodynamic theory (which assumes the
amplitude of body oscillations to be small compared with the body size), such
oscillations are unrealistically large (except in calm seas, say H s  1 m) and so are the
values of absorbed power. Of course, this is also true in general, whenever the theory
predicts large oscillation amplitudes as a result of a wave energy converter being tuned
(by phase control or otherwise) to the incoming waves.
36
10  f e (MN)
d
dt  H s
 Hs
Fig. 3.26. As in Fig. 3.25, for R  16, G  4.2  106 s kg . P H s2  28.5 kW m 2 .
It should be noted that values of the latching control parameter R much larger than
unity (required to maximize P ) may imply very large forces to keep the body fixed
prior to its release. Such forces are likely to exceed the practical limits of the ram and
remaining hydraulic circuit and would possibly require a special braking system. This is
an engineering problem that has to be faced whenever phase control by latching is
considered.
Figure 3.26 shows that the peaks of velocity d dt in general (but not in every
oscillation) coincide, in time, approximately with the peaks of the excitation force f e ,
which matches the optimal condition expressed by Eq. (3.32).
The values of P H s2 ,  H s and  H s plotted in Figs. 3.22 to 3.26 were computed
for H s  2 m. These values would change with H s due to the nonlinear response of the
gas accumulator. Provided the accumulator is appropriately sized, those changes are
relatively small within the range of H s in which the linear wave theory is applicable. If
this is the case, one can say approximately that the latching control algorithm proposed
and numerically optimized here is approximately independent of significant wave
height.
One should bear in mind that the pressure difference p decreases (due to the
continuous flow of liquid from the HP to the LP reservoir through the hydraulic
machine) whenever the floater in unable to move (this decrease is faster the smaller the
accumulator size); this effect tends to adjust the pressure level p to the current sea
state and also (in a different measure) to the wave group or even the wave-to-wave
37
succession. Naturally, the choice of the size and other specifications of the accumulator
are dictated by several criteria, namely the maximum allowable working pressure, the
desired power output smoothness and equipment costs.
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